Divergence Test

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Let $\sequence {a_n}$ be a sequence in $\R$.

If $\ds \lim_{k \mathop \to \infty} a_k \ne 0$, then $\ds \sum_{i \mathop = 1}^\infty a_n$ diverges.


We know that Terms in Convergent Series Converge to Zero.

This is the contrapositive statement of this theorem.

Thus, the theorem holds by Rule of Transposition.


Also known as

This theorem is also known as the $n$th Term Test. The reason for this is neither apparent nor obvious.